TIME-TO-FAILURE FORECAST FOR CORRODED SHELL OF ABOVE-GROUND STEEL TANK USED TO

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TIME-TO-FAILURE FORECAST FOR CORRODED SHELL OF ABOVE-GROUND STEEL TANK USED TO

STORE LIQUID FUELS

M. MASLAK

¹

, M. PAZDANOWSKI

²

An original simplified procedure to estimate the remaining service time of corroded shell of an on-the-ground steel tank used to store liquid fuels is presented in this paper. Current corrosion progress trend, identified a’posteriori based on the obligatory technical condition monitoring, is extrapolated to the future tank service time under the assumption that the conditions of service would not change and no renovation or modernization works would be undertaken. Failure probability understood as exhaustion of the capability to safely resist the loads applied due to the corrosion progress constitutes the measure of the sought uptime. For comparative purposes several effective inference methods have been proposed for the same input data, based on formally qualitatively different but corresponding description measures. It has been shown, that in the analysis of this type the representative values, usually expressed as quantiles of probability distributions describing random variables in use, need not be specified to verify the safety condition. The proposed algorithm is based on fully probabilistic considerations, and those, according to Authors’ opinion, by their nature lead to more reliable, and at the same time, objective estimates.

Keywords: steel tank, corroded shell, failure probability, durability prediction, time-to-failure forecast, remaining service time.

1Dr hab. Eng., prof. CUT, Cracow University of Technology, Faculty of Civil Engineering, Warszawska 24, 31-155 Cracow, Poland, e-mail: mmaslak@pk.edu.pl

2Dr Eng., Cracow University of Technology, Faculty of Civil Engineering, Warszawska 24, 31-155 Cracow, Poland, e-mail: michal@l5.pk.edu.pl

ARCHIVES OF CIVIL ENGINEERING

Vol. LXVII ISSUE 1 2021

DOI: 10.24425/ace.2021.136475

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (CC BY-NC-ND 4.0, https://creativecommons.org/licenses/by-nc-nd/4.0/), which per-mits use, distribution, and reproduction in any medium, provided that the Article is properly cited, the use is non-commercial, and no modifications or adaptations are made.

WARSAW UNIVERSITY OF TECHNOLOGY FACULTY OF CIVIL ENGINEERING

COMMITTEE FOR CIVIL AND WATER ENGINEERING POLISH ACADEMY OF SCIENCES

Index 351733

ISSN 1230-2945

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1. INTRODUCTION

On-the-ground steel tanks used to store the liquid petroleum fuels should be treated as ageing structures in the sense of reliability theory, due to the corrosion weakening of their cylindrical shells progressing in time. The basic source of corrosion risk may be attributed to the internal exposure of sheathing plates to the potential chemical reactions with aggressive media, often contaminated, and exhibiting high level of sulphur content [1-4]. The exposure to external atmospheric conditions seems to be only an additional aggravating factor [5-7]. Monitoring the intensity of corrosion weakening in this type of structures belongs to the basic tasks of service personnel managing the fuel depots and oil refineries. In practice the actions undertaken are usually limited to the periodical evaluation of technical condition, as required by the law [8], and possible consideration of the need to undertake the necessary renovation or modernization works. With such an approach the strategic planning and reasonable management of available resources in both economic as well as purely technical aspects is difficult. In the Authors’ opinion, due to the strategic importance of the monitored resources, the traditional evaluation of technical condition should be enriched by the elements of qualitative and quantitative risk analysis, combining the probabilistic procedures with calibration of potential failure consequences. Therefore an algorithm allowing to estimate the so called forecast remaining service time for a steel tank used to store the liquid petroleum products is proposed for practical application.

In our interpretation, this is the time counted beginning at the moment of the technical inspection of the considered structure until the anticipated future loss of the capability to safely resist the applied loads, of course subject to the condition, that the service regimen would remain unchanged during the whole period, and no maintenance activities resulting in the possible strengthening of the bearing structure would be undertaken. In this sense this is simply forecast durability, quantified only due to the corrosion weakening progressing in time (all the remaining factors generating additional risks in this domain are disregarded here). The estimated service time will be called by us “time-to-failure (TTF)” in the following analysis. It has to be underlined however, that the traditionally quantified in the relevant bibliography “mean-time-to-failure (MTTF)” remains beyond the scope of our interest.

Instead, the appropriate for this random variable, quantile of the probability distribution determined for the maximum acceptable failure probability level is sought. We will limit our considerations to the computational scenario, where the bearing capacity of the corroded tank shell is limited by the ability to resist the hoop tensile force. This happens, when the tank is completely filled with the stored product. The alternative scenario, dealing with local stability loss of empty tank shell will be considered in a separate work.

00$6/$.03$='$12:6.,

(3)

2. CONVENTIONAL TECHNICAL CONDITION EVALUATION OF A CORRODED TANK SHELL

A survey of the real, observed during inspection, sheathing plates thickness of the corroded tank shell degraded in service is the main task of the expert assessing the evaluated tank. After the inspection the expert has at his disposal the uniform set of N results for each shell section, which is subjected to statistical treatment. In the traditional approach, based on the available data set the expert determines empirical parameters of the random corroded plate thickness probability distribution (the parameters determined on a sample will be denoted by the superscript asterisk in the following considerations), i.e. the mean value m_t and standard deviation or alternatively coefficient of variation , as estimators of the probabilistic moments for a normal probability distribution

:

(2.1)

1

1 ^N

t i

i

m t

N

¦

^,

²

1

1 ^N

t i t

i

t m

V N

¦

^and ^t ^t

t

v m V

.

Based on this, the design value of the random thickness representative for the considered shell section is equal:

(2.2) ^t^d ^m^t³^V^t ^m^t

^{1 3} ^v^t

^.

At the moment of the technical inspection, further denoted as (the moment denotes in this notation the beginning of the service life), the corroded shell section is capable of safely resisting the load applied to it, provided that the below inequality is satisfied:

(2.3) ^t^d ^t^d

^{W W}

^t^max

^t^{( )}^d^e^,^t^d^{( )}^w

^.

In the above the thickness is due to the service condition:

Vt

vt

N m_t,V_t

td

W!W0 W0

( )e

td

7,0(72)$,/85()25(&$67)25&2552'('6+(//2)$%29(*5281'67((/

(4)

(2.4)

^{F p}^, ^p ^red ^{F E}^, ^E^e

^e _M^y_,0

d

r f

H p

J U J t

J

§ ·

¨¨© ¸¸¹d ,

while the thickness is due to the water test condition:

(2.5)

^{F w}^, ^w ^red ^{F E}^, ^E^w

^w _M^y_,0

d

r f

H p

J U J t J

§ ·

¨¨© ¸¸¹d ,

where defines the shell depth at which the bearing capacity condition is verified, measured relative to the stored fuel surface at the completely filled tank. The quantities and represent the overpressure generated in the gas zone of the tank equipped with permanent roof, or alternatively the dead weight of the floating roof divided by the cross sectional area of the tank equipped with floating roof, while and represent specific gravity of the fuel stored in the tank and water used in the water test (Fig. 1). Additionally denotes the radius of the tank measured with respect to the central axis of the shell while denotes yield limit of the steel the tank has been made of. Symbols and denote partial safety factors typical for the traditional approach to the standard limit states method.

Fig 1. Tank load determining the required shell thickness: at left hydrostatic pressure, at right overpressure.

In the Authors’ opinion the quantitative difference between the design value of the strength specified for the steel used to make the tank shell plates at the moment of acceptance for service and at the time of testing its technical condition after many years in service should be taken into consideration. For the moment this strength is usually determined by the following formula:

( )w

td

Hred

( )e

pE ª¬kN m²º¼

( )w

pE ª¬kN m²º¼

Up ª¬^{kN m}³º¼ U^w ª¬^{kN m}³º¼

> @

^m

r

> @

y MPa

f JF JM

fd

W0

W!W0

W0

00$6/$.03$='$12:6.,

(5)

(2.6) ^f^d

^W⁰ ^P^y^exp

³ ^{X X}^R² ²^A

^,

where and represent the median value and logarithmic coefficient of variation of the steel yield limit, respectively, while - logarithmic coefficient of variation of the element cross section and influence of imperfections (in the case of tank shell plates this pertains in general to the initial variation in plate thicknesses due to the acceptable mill tolerances). Based on statistical research [9],

for structural steels made in Poland and , resulting in . The

design value of the strength estimated with an assumed safety margin for the whole population of plates of this type is by its nature lower than the strength related to a specific implementation. The empirical coefficient of variation v_tdetermined according to (2.1) covers the influence of real (and thus smaller than the previously conservatively estimated for quantification according to the standard) initial variability . This is superimposed over the variability generated during the service by the random nature of the corrosion process. Thus the design value of bearing capacity of the shell thick should be determined based on the adjusted value:

(2.7) ^f^d ^f^d

^W ^P^y^exp

³^X^R

^.

The median and logarithmic coefficient of variation in this case should be determined based on the real steel yield limit measurements, performed at various locations on the shell simultaneously with the measurements of its random thickness. However, such action on the tanks used to store liquid petroleum products may be undertaken rather infrequently, and in general only after emptying, therefore usually only code values and are applied to evaluate the bearing capacity of the tank during inspection. Replacement of the value by implies a modification of the safety condition (2.3). A comparison of bearing capacities yields:

(2.8) ^d ^max

^d^{( )}^e^, ^d^{( )}^w

^d

⁰

d

t t t f f

W

t .

Py XR

XA

0, 08

XR XA ^{0, 06} X XR² A² ^0,10

⁰

fd W

⁰

A A

v |X W

td

Py XR

y y

P P XR XR

fd fd

^{( )} ^{( )}

⁰

max ^e, ^w

d d d d d

t f t t t f W

7,0(72)$,/85()25(&$67)25&2552'('6+(//2)$%29(*5281'67((/

(6)

In addition, the experimental research reported in [10-12] indicated, that due to the corrosion progressing in the extended service time of the tank the median value itself decreases as well.

This is a result of intensifying degradation changes of various nature weakening the integrity of ferrite grains in the corroded steel microstructure. Unequivocal quantification of such phenomena is so far difficult to estimate, as it requires much wider experimental basis. A certain modification of the approach presented here has been proposed in [13]. A relative corrosion loss related to the nominal plate thickness is assumed, subject to the assumption, that at the moment the equality

was satisfied, as due to the mill tolerances the real initial plate thickness could have been higher or lower than the nominal one with equal probability. This is basically an approximation, as the plates rolled in the 1970’s and 1980’s generally had negative tolerances. A random quotient

nom i i t t

c is a measure of the observed loss in this approach. The probability distribution parameters of the variable c are estimated by determining the median for each shell layer and empirical variation adjusted for the small sample. This results in:

(2.9)

1

ln 1 ln

N

c i c

i

N c

P

¦

oP ^and ^c ¹₁_i^N₁^ln² ⁱ

c

X N P

§ ·

¨ ¸

¦

The bearing capacity condition specified for the moment in this case should be verified for the nominal plate thickness , at the design value of the strength reduced according to the formula:

(2.10) ^f^d ^{P P}^y ^c^exp©¨^§³ ^X^R²

^X^c ²^·¸¹ .

The main advantage of the above presented approaches, which may be applied to estimate the bearing capacity of a corroded tank shell lies in their simplicity. However, in each case calculations of this type yield only approximate results. In the following considerations we propose to replace this simplified approach by an alternative one, based on the fully probabilistic inference procedures.

W Py

tnom W₀

⁰

nom t

t m W

Pc

^X^c ²

W

tnom f_d

00$6/$.03$='$12:6.,

(7)

3. RANDOM BEARING CAPACITY AND RANDOM LOADS OF TANK SHELL

The tank shell is a thin cylindrical shell of random thickness and non-random radius r (the potential geometrical imperfections are not being considered here). In current analysis it is assumed, that the bending moments acting on the shell are negligibly small and may be disregarded. In such case the hoop tensile force is authoritative for safety estimates. This force is induced in the shell plates by the simultaneous action of the hydrostatic pressure and overpressure . Its value may be expressed as:

(3.1) ^N^M

^U^p^H^red^p^E

^r^,

In the random implementation it may not exceed the bearing capacity:

(3.2) N_R f t_y .

In the following considerations it is assumed that and are mutually independent random variables (in reality the steel yield limit is inversely related to the plate thickness). Under the assumption that both thickness and yield strength are described by the log-normal probability distributions having the parameters and , respectively, the stability properties of the distributions of this type with respect to multiplication imply that the variable N_Ras well is characterized by the log-normal probability distribution having the parameters . At

occurs:

(3.3) P_NR P P_y_t ^and ^X^NR ^X²^R

^X^t ²^,

where:

t W

N_M

Up pE

N_M N_R

^t^, ^t

N P X ^N

^{P X}^y^, ^R

^NR^, ^NR

N P X W W

7,0(72)$,/85()25(&$67)25&2552'('6+(//2)$%29(*5281'67((/

(8)

(3.4)

1

ln 1 ln

N

t i t

i

N t

P

¦

oP ^and ^t ¹₁_i^N₁^ln² ⁱ

t

t X N

P

§ ·

¨ ¸

¦

The probability distribution parameters of random force are determined based on the values of partial safety factors available in the design codes, subject to the assumption that the mean values of these loads correspond to their characteristic values. Thus when the random loads , and are described by normal probability distributions the following holds:

(3.5) U_{p k}_, m_p ^and ^U^{p d}^, ^{U J}^{p k}^, ^{F p}^, ^m^p

^{1 3} ^Q^p

^o^Q^p ^J^{F p}^,₃¹^,

(3.6) U_{w k}_, m_w ^and ^, ^, ^,

^{1 3}

^, ¹

3

F w

w d w k F w mw w w J

U U J Q oQ ,

(3.7) p_{E k}_, m_E and ^p^{E d}^, ^p^{E k}^,^J^{F E}^, ^m^E

^{1 3} ^Q^E

^o^Q^E ^J^{F E}^,₃¹^.

Since random loads , and are characterized by normal probability distribution and the random force is a linear combination of these, its distribution is normal as well, and therefore:

(3.8) ^m^N^{( )}^e^M

^{m z}^p ^m^E

^r^, ^V^N^{( )}^e^M ^r

^V^p^H^red

²^VÊ² ând ^Q^{( )}^Nê^M ^Q^p²^vÊ²^,

(3.9) ^m^N^{( )}^wM

^{m z m}^w ^E

^r, V^N^{( )}^wM ^r

V^w^H^red

²V^E² and Q_N^{( )}^w_M Q_w²v_E² .

The representative design value of bearing capacity is defined in paper [14] subject to the assumed value of material coefficient as:

(3.10) ^{R d}^, ^NR

^{1 1, 645} ^NR

m

N m Q

J

.

N_M

Up Uw pE

N_M ^N

^m^N^M^,^Q^N^I

,

NR d

Jm

00$6/$.03$='$12:6.,

(9)

When the log-normal probability distribution is applied to model the random bearing capacity of a plate the formula (3.10) is reduced to the following:

(3.11) ^{R d}^, ^NR^exp

^{1, 645} ^NR

m

N PJ X

.

Definition of the design value is reasonable only when it is accompanied by the definition of the random load design value against which it could be compared. Should one assume, that the parameters of the random load are time independent, one may also assume that and

. For the quantities determined according to (3.8) and (3.9) the following holds:

(3.12) ^N^M^{( )}^e^,^d ^m^{( )}^N^e^M

^{1 3} ^Q^N^{( )}^e^M

^and ^N^M^{( )}^w^,^d ^m^N^{( )}^w^M

^{1 3} ^Q^N^{( )}^w^M

^.

The safe service condition requires that . Therefore the global factor defined below may be treated as the measure of safety.

(3.13) _{( )}^, _{( )}

, ,

max( , ) 1

R d

e w

d d

N N_M N_M

J t .

An application of the representative values and in the analysis in our opinion constitutes an unnecessary simplification of the computational model.

4. FAILURE PROBABILITY ESTIMATED FOR THE MOMENT OF TECHNICAL INSPECTION

The method proposed by the Authors to infer on the safety level of the corroded tank shell at the moment of technical inspection is based on the estimated failure probability of this shell, understood as the loss of the capacity to safely resist the loads applied to it. Let the be the random safety margin related to the moment . The value is equivalent to safe service condition, while denotes failure. The difference in this formulation represents a new random variable

,

NR d

N_M,d

N N

m_M m _M

NM NM

Q Q

,d R d,

N_M N

,

NR d ^N^M^,^d ^max

^N^M^{( )}^,^e^d^,^N^M^{( )}^w^,^d

NR N_M

'

W ' ! ⁰

0

' d '

7,0(72)$,/85()25(&$67)25&2552'('6+(//2)$%29(*5281'67((/

(10)

being a linear combination of two variables and characterized by normal probability distributions. Therefore it is described by a normal probability distribution as well. In this convention the failure probability is determined by the formula:

(4.1) ^:^P

^N^R^*^d^N^M

^P ^{' d}^* ⁰

^F ^' ^,

where is a cumulative distribution function of the random variable , while the probability of fail-safe service of the shell is its complement:

(4.2) ^R ^:^{1- 0}^P

^N^R^!^N^M

^P ^{' !}

¹ ^F

^' ^.

Since the random safety margin is described by the normal probability distribution:

(4.3)

²

0

1 1

0 exp

2 2

m d

SV V

f

'

' '

ª §' · º

« »

' ' ! ¨ ¸ '

« © ¹ »

¬ ¼

³

R P .

After introduction of the standardized random variable one obtains:

(4.4)

2

1 exp 1

2 _u 2

u u du u

S

f

§ ·

¨ ¸ )

¨ ¸

© ¹

³

R .

So at the same time:

(4.5)

2

1 exp

2 2

u u

u du u

S

f

§ ·

¨ ¸

: )

¨ ¸

© ¹

³

^.

Function is the cumulative distribution function of the normal probability distribution of a standardized random variable , i.e. the Laplace function available in statistical tables. The failure

NR N_M

N m',Q'

F ' '

'

u m V

'

^u

)

u

00$6/$.03$='$12:6.,

(11)

probability estimated by the evaluator should not exceed the admissible level , i.e. the highest level acceptable by the tank user. This level corresponds to the quantile of the variable set

at the level of , meaning that . Then, by definition . In the

following considerations is treated as a positive parameter by assumption. For such a convention the following holds:

(4.6) ^{: :}^ult

^u^req⁰

^:¹

^u^req^!⁰

^:

^u^req^!⁰

^.

When the probability value is known, one may determine the required limit value , or alternatively (the symbol inv) denotes the inverse Laplace function here).

(4.7) ^{: )}^ult

^u^req

ôû^req ^{) :}înv

^ult

^,

(4.8) _req ^req m _req _req

u m u V

V

'

' '

'

' o ' .

The failure will occur when the random bearing capacity , decreasing with progressing corrosion, will get equal to the random load . Then , thus for :

(4.9)

0

2 2

0 NR N

NR N

m m

u u m ^M

V V V M

'

.

The value determined in such manner may be interpreted as the global safety index combining the partial safety indices and . Should one keep in force the convention, that both

and are positive, the safety condition would take the following form:

(4.10) ^{: :}

^u⁰ ^{: :}^ult

^u^req

^,

^u

: :ult

u

u ureq ^{: :}^ult

^u ^u^req

^u^req⁰

ureq

:ult u_req!0

req

'

NR

N_M ' 0 u!0

u0 E'

R NR

E E ES ENM

u u_ult

7,0(72)$,/85()25(&$67)25&2552'('6+(//2)$%29(*5281'67((/

(12)

which is equivalent to the formulation:

(4.11) u₀!u_req and ' ! ' . 0 _req

The condition (4.10) may be transformed [15] by introducing the average (central) safety factor into the analysis, such that:

(4.12) ^NR _req

N

m m_M

J tJ .

Then, by (4.9) it follows, that:

(4.13)

0 2

2

1

NR N

u

v v_M J J

.

The necessary value of is determined directly from (4.13), by taking , which in turn leads to:

(4.14)

2 2 2 2

2 2

1 1 1 1

1

req NR req N

req

req NR

u v u v

u v J M

.

5. CORROSION PROCESS PROGRESS FORECAST

The basic requirement for forecasting future changes in the safety condition of the corroded shell of the tank is the most reliable extrapolation of the trend describing the corrosion progress to date for the duration of the facility's further service. It is also assumed, that the service mode and other parameters such as climate conditions, air pollution etc., however with random instantaneous values, in the longer period of time would not change (may be described by a narrowband, stationary random process). In current considerations it is assumed, that the decrease in average shell plate thickness due to corrosion losses may be described by the following linear function with a sufficiently high precision:

J Jreq u0 ureq

00$6/$.03$='$12:6.,

(13)

(5.1) ^m^t

^W ^m^t

^W⁰ ^A^W^,

where is the mean thickness of the analyzed plate forecast for the time , - the mean thickness of the plate observed (hypothetically) at the moment of tank entering service, - directional coefficient of the trend line modelling corrosion progress, averaged over measurements.

There are no formal obstacles to assuming various nonlinear trends [16-18] in a more detailed analysis. So, would the parameters in the formula (5.1) be calibrated to obtain for the moment in time

the corrosion state observed at the moment of measurements, one could unequivocally determine the value of the directional coefficient A . In the corrosion progress model assumed here this value remains constant for the whole tank service time. It is assumed that at the moment of tank entering service , and this in turn means, that . Based on the preceding:

(5.2) t^nom m^t

A W

.

After entering (5.2) into (5.1) for one obtains:

(5.3) ^m^t

^W ^m^t_W^W ^t^nom§©¨¹_W^W ·¸¹ , ^{V W}^t

|^V^t¨ ¸© ¹§ ·_W^W and

₁ ₁

t t

t

t nom

t

v

m t

m Q W V W

W W

W

§ ·

¨ ¸

© ¹

.

The formula for applied here is simplified, as the influence of the initial variability of plate shell thickness on has been disregarded. The component takes this influence into account for . The exact formula would also require the knowledge of the coefficient of variance , which is usually unknown and difficult to obtain at the moment , as well as taking into account the mutual relationship between and .

mt W W ^m^t

W⁰

A

W

0

t nom

m W t mt tnomAW

W W!

V Wt

⁰

t W ^{V W W}^t

^!

^V^t

W Wd

⁰

vt W W

t ^t

^W⁰

7,0(72)$,/85()25(&$67)25&2552'('6+(//2)$%29(*5281'67((/

(14)

6. FORECAST OF FUTURE CHANGES IN FAILURE PROBABILITY

Knowledge of the formulae (5.1) and (5.3) allows for determination of the random bearing capacity parameters forecast for the assumed time . Should one opt for disregarding an insignificant difference between the median value and the mean value , then based on (3.3):

(6.1) ^m^NR

^W ^{m m}^y ^t

^W ^and^v^NR

^W ^| ^Q^R²

^v^t

^W

²^.

Random bearing capacity safety margin ' then becomes a function of time W^:

(6.2) '

W ^N^R

W ^NM

W .

This pertains to its mean value and standard deviation as well (based on the assumption, on the loading

process stationarity and ):

(6.3) ^m_'

W ^m^NR

W ^m^NM and ^{V W}^'

^{V W}^NR

²^V^NM² ^,

where and . Thus, after standardization of the random

variable :

(6.4)

u W WV W' m^' W

'

.

The probability is determined as the cumulative distribution function of the normal probability distribution of the variable , i.e. the so called Laplace function ( is the cumulative distribution function of the normal probability distribution of the variable ):

W W!

Py my

⁰

N N N

m_M W m _M W m _M V^NM

W V^NM

NR mNR vNR

V W W W VN_M m vN_M N_M

^W

'

^{W W}

: !

u W W! ^F

^'

^W

'

00$6/$.03$='$12:6.,

(15)

(6.5) ^:

^W ^P

^N^R

^W ^d^N^M

^W

^P

^'

^W ^d⁰

^F

^'

^W

⁾

^u

^W

^.

The bearing capacity of the corroded shell is exhausted for , as then the random bearing capacity gets equal to the random load . This in turn means that , and thus at the same time . Should one assume the convention, that the parameter is by default positive, then:

(6.6) ^:

^u⁰

^W

^:

^u⁰

^W ⁰

^:¹

^u⁰

^W ^!⁰

^:

^u⁰

^W ^!⁰

^,

and the equality (6.5) should be modified to the following:

(6.7) ^:

^u⁰

^W

⁾

^u⁰

^W

^.

Parameter is determined based on (6.4):

(6.8)

0 2 2

0 _NR _N

NR N

m m

u m ^M

M

W W W

V W V W V

' '

.

The safe service condition of the corroded tank shell may be expressed as satisfaction of the inequality [19-22]:

(6.9) ^:

^u⁰

^W

^{d : .}^ult

Specification of the highest failure probability acceptable to the tank user

allows for the determination of the required limit value , and subsequently (inv) stands for the inverse Laplace function here):

(6.10) ^{: )}^ult

^u^req

ôû^req ^{) :}înv

^ult

^,

0

u W u W

NR W ^NM

W '

^W ⁰ ^m'

^W

0 0

u W ^u⁰

^W

u0 W

⁰

ult u W uult

: :

req 0

u ! ^'^req

^W

7,0(72)$,/85()25(&$67)25&2552'('6+(//2)$%29(*5281'67((/

(16)

(6.11)

req

req req req

u W m W m u

W W V W

V W

'

' '

'

o ' .

The limit condition (6.9) is thus equivalent to:

(6.12) ^u⁰

^W ^!^u^req ^and ^'

^W ^{! '}⁰ ^req

^W ^.

Should one denote by the average (central) safety factor, such as:

(6.13)

^NR

^req

N

m m_M

J W W !J W ,

then the formula (6.8) would take the form:

(6.14)

0 2 2 2

1

NR N

u

v v _M

W J W

J W W

The required value is obtained by entering into the formula (6.14), this yields:

(6.15)

2 2 2 2

2 2

1 1 1 1

1

req NR req N

req

req NR

u v u v

u v W M

J W

W

.

The computational procedure presented by the Authors in this paper has been numerically verified on the example of a forecast prepared for an existing steel tank for fuel storage equipped with floating roof, located in one of fuel depots in the south of Poland. Detailed results of this analysis have been published at first in the limited scope in [23] and subsequently, after updating and adaptation to current guidelines and design standards in [24].

J W

Jreq W ^u⁰

^W ^u^req

00$6/$.03$='$12:6.,

(17)

7. CONCLUDING REMARKS

In the Authors’ opinion the proposed algorithm may be applied to effectively estimate the forecast durability of a corrosion degraded shell of a steel tank used to store liquid petroleum products. These calculations are based on the data obtained during random plate thickness measurements , performed on the shell section authoritative for the evaluation of bearing capacity of the whole shell at the moment associated with the obligatory inspection of the tank condition. Determination of the future failure-free service time under assumed, user accepted, allowable failure probability is the purpose of this analysis. The safety condition (or alternatively

) may be replaced by the condition [15], which is not necessarily more convenient to use. The proposed approach is based on the assumption on the stationarity of the loading process. This means, that both the mean value of the longitudinal tensile force in the shell and its statistical variability remain constant during the whole service time. However, random fluctuations of the momentary values of are not excluded. The progressing in time corrosive weakening of the shell bearing capacity is described by the mean bearing capacity monotonously decreasing in time , and associated with decreasing thickness . In this approach random fluctuations of the bearing capacity are not authoritative for the evaluation of durability. The changes in the random bearing capacity induce simultaneous increase in the coefficient of variation

. The decreasing in time values of the factor or corresponding parameter may constitute a measure of the progressing degradation of the shell section. These values are compared against required limit values or , respectively. The latter are specified for the failure probability acceptable to the tank user. A constant required level of safety during the whole tank service time is postulated, this comes to the requirement that . Simultaneously this means, that the coefficient is time independent. This statement, however, is not reuse in the case of the factor , which in order to keep the probability constant has to grow with the progressing degradation of the corroded tank shell. The forecast corrosive durability of the shell is determined by the difference of times . The time follows from

inequality , which is equivalent to the condition . The

t

W Wd

:ult ^u⁰

^{W W}^!

^!^u^req

⁰

req W

' ^{J W}

^!^J^req

^W

mN_M

vN_M

N_M

W W! ^t ^t

^W

NR NR

v v W ^{J W}

^u⁰

^W

Jreq W ^u^req

W :ult

ult const :

ureq const W W

Jreq W :ult

1,d d

W W W Wd

^d

^req

^d

J W W J W W ^u⁰

^{W W}^d

^u^req

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(18)

durability of the weakest tank shell section determined by the method described above will be authoritative for the whole tank shell (thus one deals here with a serial system in the sense of the reliability theory). The value in the traditional approach is interpreted as the global safety index . Thus a value of , such that , should be assigned to the required value of . It is usually postulated to assume a uniform value of E_'_,_req 3.8, corresponding to the value of^{: :}^ult

^3.8

^{7.237 10} ⁵. The safe values have to be smaller than that. The specification of constant value results in time dependence of partial safety indices related to the load and bearing capacity . This is because should one accept the division rule, where the so called sensitivity coefficients and such as:

(7.1) and ,

are the coefficients of proportionality, as authoritative for further considerations, then the following holds:

(7.2) .

With the progressing corrosion degradation the influence of the index increases at the expense of . Thence the conclusion, that the assignment in the conventional, code based approach to the forecast, based on the specification of the representative design values of bearing capacity and load , of uniform and service time W independent values seems to be not entirely justified.

REFERENCES

1. L. Garverick, “Corrosion in the petrochemical industry”, American Society for Metals, 1994.

2. A. Rim-Rukeh, P.A. Okokoyo, “Underside corrosion of above ground storage tanks (ASTs)”, Journal of Applied Sciences and Environmental Management 9 (1): 161-163, 2005.

3. J. Harston, F. Ropital, “Corrosion in refineries”, CRC Press Inc., 2007.

4. A. Groysman, “Corrosion problems and solutions in oil, gas, refining and petrochemical industry”, Koroze a Ochrana Materiálu 61 (3): 100-117, 2017.

u0 W W!

E' E',req E'tE',req

ureq

^u⁰

^W

: :

E',req

NM S

E E ENR ER

D WSR ^{D W}^RS

² ²

N SR

N NR

M M

D W Q

Q Q W

2 2

NR RS

NM NR

Q W D W Q Q W

,req SR S RS R

E' D W E D W E

ER

ES

,

NR d

N_M,d E_R E_S 3

00$6/$.03$='$12:6.,

(19)

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15. M. Maślak, J. Siudut, „Oszacowanie degradacji nośności losowo skorodowanego pasa płaszcza stalowego zbiornika walcowego o osi pionowej”, Ochrona przed Korozją 12: 456-461, 2007.

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20. M. Maślak, J. Siudut, “Durability prediction of randomly corroded side surface sheet in steel on the ground tank for liquid fuel storage”, Proceedings of the 4^thInternational Conference “Recent Advances in Integrity – Reliability – Failure (IRF)”, Funchal, Madeira, Portugal, June 23-27, abstract 159-160 + CD 8 p., 2013.

21. M. Maślak, M. Pazdanowski, J. Siudut, K. Tarsa, “Probability-based durability prediction for corroded shell of steel cylindrical tank for liquid fuel storage”, Proceedings of the 1^stWorkshop of COST Action TU1402:

“Quantifying the Value of Structural Health Monitoring”, DTU Civil Engineering Report: R-336, Lyngby, Denmark, May 4-5, 82-95, 2015.

22. M. Maślak, M. Pazdanowski, J. Siudut, K. Tarsa “Corrosion durability estimation for steel shell of a tank used to store liquid fuels”, Proceedings of the 12^thInternational Conference “Modern Building Materials, Structures and Techniques (MBMST)”, Vilnius, Lithuania, May 26-27, 2016, Procedia Engineering, 172, 723-730, 2017.

23. M. Maślak., J. Siudut, „Szacowanie degradacji nośności i prognozowanej trwałości losowo skorodowanego pasa płaszcza stalowego zbiornika walcowego o osi pionowej – przykład obliczeniowy”, Ochrona przed Korozją 2:

50-54, 2008.

24. M. Maślak, M. Pazdanowski. „Probability-based remaining service time prediction for corroded shell of a steel tank used for liquid fuel storage”, Proceedings of the 29^thEuropean Safety and Reliability Conference „ESREL 2019”, September 22-26, 2019, Hannover, Germany, Research Publishing Services, Singapore, 3232-3239, 2019.

7,0(72)$,/85()25(&$67)25&2552'('6+(//2)$%29(*5281'67((/